Let B be a uniformly convex Banach space whose dual space \(B^*\) is also uniformly convex. Typical example of such space is \(L_ p(G)\) \((1<p<\infty)\) where G is a domain in Euclidean space.
Let \(\phi\) be a function from \(R_+\) into \(R_+\), where \(R_+\) is the space of non-negative real numbers, strictly increasing with \(\phi (0)=0\) and \(\phi\) (r)\(\uparrow \infty\) as \(r\uparrow \infty\). \(\phi\) is called a standard function (Eichfunktion).
The dual map B into \(B^*\) induced by \(\phi\) is given by \[ J_{\phi}(b)=\{b^*\in B^*;\quad \| b^*\| =\phi (\| b\|),\quad (b^*,b)=\| b^*\| \| b\| \}. \] Furthermore, \(J_{\phi}(b)=\Psi '(\| b\|)\) with \(\Psi (r)=\int^{r}_{0}\phi (s)ds\), \(r\geq 0\), i.e \(J_{\phi}\) is a gradient of the functional and so it is recognized as potential operator.
The author of this note considered the properties of such operators and got several lemmas and applications to \(L_ p(G)\) or Sobolev spaces. He considers also other functionals defined on a subspace of Sobolev spaces whose gradient exists and has remarkable proerties such as uniform monotoneness.